The excerpt is from "Topology Without Tears" by Sidney Morris. I am not able to get the rationale for the steps (1) and (2) in the above picture. Why does "[0,1] = [0,1]$\cap$Y " imply that [0,1] is closed in (Y,$T_1$) ?
And why "[0,1]=(-1,1.5)$\cap$Y" imply [0,1] is open in (Y,$T_1$) ?
My reasoning for why [0,1] is clopen is as follows. Since (Y,$T_1$) = [0,1]$\cup$[2,3], by definition [0,1] is in this topology, hence it is open. Since it is the complement of [2,3] in Y, it is closed. Why taking an intersection with arbitrary sets is required as in the above picture? What have I misunderstood here?
Best Answer
I don't understand your argument "Since $(Y,T_1)=[0,1]\cup[2,3]$ by definition $[0,1]$ is in this topology, hence it is open". What you need here is to know the definition of topological subspace. If $(X,T)$ is a topological space and $A\subset X$ then you can define a topology on $A$ like this: $T_A=\{A\cap U: U\in T\}$. And then $(A,T_A)$ is called a subspace of $(X,T)$. So in your case the open sets in $Y$ are the sets which can be written as an intersection of an open set in $\mathbb{R}$ (with the standard topology) with $Y$. Also it is easy to prove that the closed sets in $Y$ are the sets which can be written as an intersection of a closed set in $\mathbb{R}$ with $Y$.